Two-fermion system wave function

[PhyPsy]

For a two-electron atom, this book says that the overall wave function is either a) the symmetric space function times the antisymmetric spin function or b) the antisymmetric space function times the symmetric spin function. However, in another problem which involves two fermions in a harmonic oscillator, it says the overall wave function is a sum of the two aforementioned products (a + b). Why would the atom's wave function not be a + b instead of being just a or b?

 

[mfb]

If your two-electron atom does not have a superposition of different energy levels for the electrons (something you have to carefully prepare), the spins are fixed and antisymmetric OR symmetric, but not both, and the same is true for the spatial component. If you have superpositions of different energy levels, you might get something like "a+b".

 

[PhyPsy]

For the first excited state of the atom, one electron is in the lowest energy level n=1, and the other is in the level n=2. According to your statement, there should be a superposition, but the answer only contains a b (from my initial post) term and no a term.

 

[mfb]

For the first excited state of the atom, one electron is in the lowest energy level n=1, and the other is in the level n=2.

If you fix the energy levels like this, you do not have a superposition of different energy states.

What I meant is something like Rabi oscillations, where the electrons are not in specific energy levels (at least not all the time).

 

[PhyPsy]

OK, so then for systems that are not a superposition, how do you know whether to use a or b? I understand that for atoms where both electrons have the same value for n, the antisymmetric space function is equal to 0, so you have to pick a, but what about atoms where the electrons have different values for n? Is it always b for atoms with electrons that have different values for n?

 

[cgk]

In the case of electrons, usually the antisymmetric spatial wave function is preferrable, since this leads to a larger mean separation of the electrons (look up the Hund rules).

But both are valid wave functions, and which one is preferable depends on the circumstances. For example, most molecules have singlet ground states, but then there is oxygen2, which does not.